#8701680

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4145 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,6,7)(3,8,5,4)(26,27)(28,35)(32,34)(36,37);(1,3,6,5)(2,4,7,8)(9,10,15,14)(11,19,21,16)(12,22,20,17)(13,18,23,24)(25,26,30,27)(28,31,35,33)(29,32,38,34)(36,39,37,40);(9,11,15,21)(10,16,14,19)(12,13,20,23)(17,18,22,24)(25,27,30,26)(28,33,35,31)(29,34,38,32)(36,40,37,39);(2,7)(3,5)(9,12,15,20)(10,17,14,22)(11,13,21,23)(16,18,19,24)(25,28)(26,31)(27,33)(29,37)(30,35)(32,40)(34,39)(36,38);(1,4,6,8)(2,3,7,5)(9,13)(10,18)(11,20)(12,21)(14,24)(15,23)(16,22)(17,19)(25,28)(26,31)(27,33)(29,37)(30,35)(32,40)(34,39)(36,38);(1,5,6,3)(2,4,7,8)(9,14,15,10)(11,19,21,16)(12,17,20,22)(13,18,23,24)(25,29,30,38)(26,32,27,34)(28,36,35,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8608876);